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Publication 00-CNA-09

Relaxation results in micromagnetics

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon Universit
Pittsburgh, PA 15213
email: fonseca@cmu.edu

Giovanni Leoni
Dipartimento di Scienze e Tecnologie Avanzate
Università del Piemonte
Orientale, Alessandria, Italy 15100
email: leoni@unipmn.it


Abstract:

In this paper an integral representation formula is obtained for the relaxed energy of a large ferromagnetic body. With $\Omega\subset\mathbb{R} ^{N}$ being the reference configuration of the body and $m:\Omega\rightarrow S^{N-1}$ the magnetization, consider the energy

\begin{displaymath}F(m):=\int_{\mathbb{R} ^{N}}f(x,\chi_{\Omega}(x)\,m(x),u(x),\nabla u(x))\,dx \end{displaymath}

where $(\chi_{\Omega}\,m,\nabla u)$ satisfies Maxwell's equations, i.e. $u\in H^{1}(\mathbb{R} ^{N})$ is the unique solution of $\Delta u+\mathrm{div}% (\chi_{\Omega}\,m)=0$ in $\mathbb{R} ^{N}$. If f is a Carathéodory function satisfying very mild growth conditions then it is shown that the relaxation of F with respect to $L^{\infty}$-$w\ast$ convergence in $L^{\infty}(\Omega;\overline{B(0,1)})$ is given by

\begin{displaymath}\mathcal{F}(m)=\int_{\Omega}Q_{M}\,f(x,m(x),u(x),\nabla u(x))... ...t _{\mathbb{R} ^{N}\setminus\Omega}f(x,0,u(x),\nabla u(x))\,dx \end{displaymath}

where $Q_{M}\,f$ is the quasiconvex envelope of f relative to the underlying partial differential equations. This class of integrands includes those of the type

\begin{displaymath}f(x,m,u,h)=\varphi(x,m,u)+\psi(x,u,h) \end{displaymath}

with $\psi(x,u,\cdot)$ non convex, thus extending the available relaxation results in micromagnetics.

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